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Friday, August 29, 2008

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Don't forget our Mathanagram for Aug-Sept. so far we have received a couple of correct responses. You are encouraged to make a conjecture!Look here for directions. Here is the anagram again:

PRINCE? NAH! E-ROI!

Now for discussion of the title of this post...
The English language has many confusing phrases but "as many as" IMO has blighted the youth of many an algebra student. Perhaps you think I'm exaggerating this? At the beginning of the school year, write the phrase in the title of this post on the board and have your PreAlgebra/Algebra I (or higher) students write one of the two equations on their paper. Give them only a few seconds, then compile the results. Let us know if the vast majority choose the correct equation. Of course, the outcome depends on the group and many other factors but if we have enough data it might prove interesting. I'm basing this on many years of questioning students. Perhaps I am the only one who has experienced this phenomenon!

The abstraction of algebra is difficult enough for some youngsters. Students who are new to our language have particular difficulty with idiomatic phrases but those born here also seem to struggle with the verbal parts of word problems - that's completely obvious to any algebra teacher of course. If only we could remove the words from a word problem!

Certainly teaching vocabulary and math terminology is an essential part of what we do as instructors. We should also hold students accountable for this vocabulary by assessing it directly.

In this post, I'm inviting readers to share some of the coping mechanisms and pedagogical strategies they use in the classroom to help students survive phrases like "as many as." What phrases seem to cause the most confusion among your students? How about "x is four less than y?"

Here is my initial offering. Let me know if you do something similar or if you feel this might be helpful (or if you vehemently disagree!).

KEY STEP: First decide from the wording of the problem if there are more girls or more boys. In fact, this should have been my original question -- not the equations! It is critical for students to be able to translate the verbal expression into a comparative relationship: Which is the larger quantity? Number of boys or number of girls? Hopefully, most youngsters would interpret the original problem to imply that there are more girls than boys. Hopefully! Ask this question first (metacognitively, students need to learn to ask themselves questions like this when they are reading).

NEXT STEP: Now the issue is where to place the "2" in the equation. Based on the key step above, we know that the number of girls is the larger quantity. Ask them why 2G = B would be incorrect.

Better alternative for some:
We all know that those who have difficulty handling abstraction benefit from concretization, i.e., using numerical values:

Have them write both possibilities:B = 2G and G = 2B
Now have them substitute values for G and B that make sense for the original problem, sayG = 12, B = 6. Some struggle with this!
By substituting (students like the phrase "plug in") these values into both equations, they should see that 6 = 2⋅12 does not make sense. The correct equation should become apparent. Should...
Of course, most youngsters need to practice many of these before they reach comfort level.

16 comments:

Agreed. They spend so much time complaining about "when will we ever use this?" Then you give them a situation and they hate it. Not only the translation of words into math, but then that their "real life" answers are not nice integers! Ahhhhh!

This is where I struggle when I teach algebra 2. It has become so second nature to me which is the right equation that I struggle to understand how they can possibly see the other way. Just reading the title here was so confusing to me that I was trying to figure out how the other one might possibly make sense (maybe if there were negative boys and girls?).

Dave,That's the same problem we all face: The correct equation seems so obvious to us, how could they not "see" it! Try imagining how you felt the first time someone handed you directions for some new technology and told you it's so easy!!I write this blog to share my own experiences and maybe help a few along the way. Thanks for sharing.

I'm tempted to compile a list of common English phrases that create confusion. Most Algebra 1 texts provide plenty of practice for translating common verbal expressions but we could always use more! Also, standardized tests (SATs, ACT, Algebra I state tests, American Diploma Project End of Course Exam in Algebra, etc.) generally include some of these.

What are some other phrases you've come across that confound childrens' brains?

I am definitely using this when our co-op classes start next Friday. I have two groups: middle school and high school. The youngers can have the "which group is bigger" hint, but I'm giving it to the older kids straight. I wonder how many of them will get it?

I think the trouble with these (and many other word problem mix-ups) is that the order of the words in the sentence is different from the equation. Students read twice ... girls ... boys and the "correct" equation seems so obvious that they don't bother to think. Similarly with Jonathan's ratio problem.

Denise--I agree that the order of the words or symbols as in Jonathan's example is a critical factor. I've been thinking that it may even go beyond this to the essential core of problem-solving. We need to train our students to think more deeply about what they're doing. The natural tendency for most students (and adults too for that matter) is to respond quickly, almost automatically, rather than take the time to read more critically and THINK about what they are doing. Pretty obvious, huh! As teachers we say these things to our students, but it often falls on deaf ears. We tend to therefore fall into the trap of 'survival mode' for our students, giving them a list of keywords or phrases to look out for. You know what I mean, Denise. "If the problem uses this word, children, then you should..."

Now, I do feel that some of this is absolutely necessary to help our students avoid common traps in the phrasing or in the math. However, I suggested in my post that we need to encourage some other constructs or frameworks for them. I was interested in your take on the "Which is the larger quantity, which is the smaller quantity?" This metacognitive approach seems to me to be more effective than merely memorizing that you must remember to reverse x and y when encountering the phrase "y less than x." Of course, in the light of day, we will all tell them to reverse it, because it is a survival tactic!

The other heuristic I mentioned was concretizing the problem, that is, training our students to replace "y less than x" by "what is 4 less than 7?" It's not enough for us to show them this. We need to train them to do this by themselves, since it may not be natural for most. Of course, some youngsters do not need these strategies. They do these things instinctively.

By the way, I use the 'reverse cross-multiply' strategy for handling the "3x = 5y, what is x/y?" type. Students write a proportion x/y = __ / __, then fill in the blanks to insure that the result is 3x = 5y when cross-multiplying. Thus the "3" must be diagonally opposite the x, etc. Yes, this is a tactic, but it does have meaning if the student understands why cross-multiplication is a valid operation. I wonder how many students really know WHY that method works!

Here are our results from last Friday:middle school--right:wrong = 3:2.high school--right:wrong = 3:5, with one person caging his or her bet by writing both A and B on top of each other.

I gave the middle school students the "Which group is bigger?" and about 10 seconds to respond. I wanted to push the high school students to give an instinctive response, so I allowed them only 3-5 seconds.

When we went over it in class, I think everyone understood how testing a few numbers can be a useful check to see if you have the equation correct. Whether they will remember to do that in future ratio problems remains to be seen.

I also tried this with my 4th grader, who naturally gave the wrong answer. Then I went over the scaffolding questions and several numerical examples with her, and she could answer all the questions correctly. But when we went back to the original, she again gave 2G=B as her answer.

She definitely understood the situation, and she was confident that the 2G=B equation expressed exactly what was going on. I think she must have been interpreting the equation as if it were a ratio: "2 girls for every boy."

By the way, the puzzle inspired her to write a blog post: Tons of girls.

Thank you Denise. It's very gratifying to see how these play out with real students. I'm not surprised by the results at all. Your daughter will eventually make sense out of the convoluted wording which is the real nub here.

Thanks Dave... I'll be using this one in my 9th grade Integrated Algebra class and I'll let you know how they do. A similar problem that most of my students have trouble with is something like "seven less than ten". They always want to write 7-10 rather than 10-7. It is one of those backwards ones. Do you know of any other backwards phrases to be aware of?

Eric--Strange that this post from some time ago still generates many visits and occasional comments. I think it's because those of us who have experienced the student confusion resulting from these kinds of convoluted or inverted verbal expressions empathize with each other! Sometimes we simply teach students survival skills as you have done: "When you see this phrase, always ....."

Subtraction and division definitely create problems because of the variety of phrases like:(1) "M less than N": N-M(2) "M less N": M-N(3) "M is less than N": M < N(4) "The difference of M and N": M-N(5) "M is three less than twice N": M = 2N-3etc...

Division Phrases:(1) "M divided by N": M/N(2) "The quotient of M and N": M/N(3) "The ratio of M to N": M/N(4) "M is what fractional part of N?": M/N(5) "M of N" (where M is a fraction): MN

These tend to cause issues because subtraction and division are not commutative so order is critical.

Then you have terms like factor and multiple. I tend to keep it concrete:3 is a factor of 1818 is a multiple of 3

I've occasionally used thhe islly mnemonic: ___F)M

Most students listen to the FM band so maybe they will remember that the "factor" (F) is on the outside of the division and the "multiple" (M) is on the inside. Then again, transferring from division form to fraction form can be an obstacle for students.

If I think of others, I'll add them. If others have their favorites, pls share them!!

Thank you for your reply and help. I definitely have to go over some of those other phrases with my students and the mnemonic will help. You seem like you have a lot of experience. This is my first year as a teacher and I am teaching 9th grade Integrated Algebra in NYC. Do you have any pointers for me? I will be starting to review for the regents exam in about a month and a half... any suggestions. What do you teach? Thanks for all your help.

Eric--Congratulations on getting through the first 75% of your 1st year!

I'm retired from the classroom at this time, although I still teach SAT classes. My first year teaching was back in '70-71 at Queens College, I went on to teach secondary math for the next 37 years. Experience helps so definitely connect with veteran teachers and ask lots of questions -- I'm sure you already do! I'm not an expert on NY state curriculum but I know there are websites which provide samples of released Regents. My compatriot, Jonathan, who blogs over at jd2718 (just Google that!) is far more knowledgeable about NYS and the integrated algebra curriculum.

I would also recommend going to the Achieve/ADP websites. The links are in the sidebar of my blog. There will soon be a link to a full practice test for first year algebra which will provide additional practice for your students. I'm guessing "integrated" means a sprinkling of discrete math topics in the algebra, perhaps with some geometry added in, or am I way off?

I am a math PhD in Canada and my mother tongue is Chinese. So far my only trouble understanding English expressions about mathematics is this "twice as many" stuff - have to look up the internet every time. Also I know that many of my friends have this same problem.

Eh, with the internet being what it has become, I expect you'll still be seeing responses in 30 years.

For myself, I'm a math-challenged adult trying to get started on an engineering degree, and having a hard time making the mental jumps from natural language to mathematics. Posts like yours are incredibly helpful.

Thanks, Bryant. Yes this post will outlive me! Maybe I should have the title as my epitaph!You'll survive the math. There are so many good online math tutorials. Also feel free to send me a question via the Blogger Contact Form at the top of the right sidebar of my home page.

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ZERO IS A 'WEIRDO'! (W)hole(E)ven(I)nteger(R)Rational/Real(DO) Cannot Divide by O!BUT Zero is NOT Positive and NOT Negative!

POSITIVE INTEGERS start from 1

PRIMES start from 2 (not 1)

INTEGERS can be NEGative (and zero!) as well as positive

MEMORIZE the formula for the nth term of an arithmetic sequence: a(n) = a(1) + (n-1)d.Example: Consider the sequence of positive integers which leave a remainder of 3 when divided by 4. What is the 100th term?Step 1: List the first few terms 3,7,11,15,... to see the pattern and recognize it is an arithmetic sequence.Step 2: Identify the values which are givenFirst term or a(1) = 3Common difference or d = 4Number of terms or position of desired term or n = 100Step 3: Substitute into formula and solvea(100) = a(1) + (100-1)(4) = 3 + (99)(4) = 399

Of course there are other ways to find the 100th term such as 100 x 4 - 1 but the formula is so useful for so many types of questions it is worth learning!

Know the above by heart and you are way ahead of the game! These facts will absolutely be needed on your next SAT or standardized test!

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Recently retired math educator and Supervisor of Mathematics; 30 years experience as an Advanced Placement Calculus (BC) teacher; Former Author of Math Teachers of New Jersey Annual HS Math Contest; Former K-5 Chair of New Jersey Math Content Standards and Curriculum Frameworks; Former member of Math Item Review Committee for New Jersey High School Proficiency Assessment; Experienced SAT Math Instructor and author of SAT materials; speaker at many regional and national math conferences